詳解：(1)設 \(f(x) = \sqrt[3]{x}\) ， \(f'(x) = \frac{1}{{3\sqrt[3]{{{x^2}}}}}\)

(2)微分公式 \(f(x + \Delta x) \buildrel\textstyle.\over= f(x) + f'(x)\Delta x\)

(3)令 \(x = 1000\) ， \(\Delta x = 3\)  \(f(1000 + 3) \buildrel\textstyle.\over= f(1000) + f'(1000) \cdot 3\)

\(\sqrt[3]{{1003}} \buildrel\textstyle.\over= 10 + \frac{1}{{3\sqrt[3]{{{{1000}^2}}}}}3 = 10.01\)

故選(B)